Why do we use the term multicollinearity, when the vectors representing two variables are never truly collinear?

When two vectors $a$ and $b$ are collinear, then $a = xb$, (where $x$ is a scalar) so in linear algebra, collinearity is a narrowly and clearly defined (and binary) concept. Two vectors -- in my understanding -- are either collinear or they are not; there are no different degrees of collinearity. If $a$ and $b$ represent distinct time series or samples in econometrics, I would therefore expect to never actually find (multi)collinearity in any empirical context, because it is close to impossible that two distinct time series or samples are exact multiples of one another. We can find correlation, and I could also imagine finding variables which are -- at least approximately -- coplanar in certain contexts, but never true collinearity.

Why then is the term multicollinearity used in the context of econometrics, when what it really seems to mean is a (statistically significant) correlation between two or more explanatory variables in a multiple regression model? Is what's meant by problems of multicollinearity that the null hypothesis of no collinearity is rejected at a certain significance level, even when we know for certain that there is strictly speaking no collinearity among the explanatory variables simply by inspecting the data? Why is multiple correlation not the more accurate term to use?

I have recently encountered the terms perfect and imperfect multicollinearity and this has confused me additionally. Does someone have a rigorous understanding of this and could share it? I would very much appreciate it!

• It's a good question. But in a comparable sense it is also the case in linear algebra that a system of linear equations $Ax=b$ has no solutions when $b$ is not in the sub-vector space spanned by the columns of $A$. Nevertheless, least squares finds solutions, but rarely "true solutions" (they would be the ones whose residuals are all zeros). – whuber Dec 11 '14 at 23:56

I don't think anyone worries about exact collinearity. If that was the case, $X'X$ would not be invertible. That is why full column rank of $X$ in one of the first assumptions. People worry about inexact relationships, since then there are coefficients to interpret, but they are too unreliable to be useful. The world multicollinearity is usually reserved for the latter. But one can easily become the other in the limit. Think about $\vert X'X\vert$. This determinant declines in value with increasing collinearity, tending to zero as the collinearity becomes exact. You can also consider the auxiliary regression $R^2_j$ tending to 1. Perhaps that is why we use the same term for both, though the extreme case is impossible if you have coefficients in hand.
There are various sample measures that you can compute and report (VIFs or conditioning indices), to help gauge how severe this problem may be. But they're not statistical tests. Because multicollinearity is a characteristic of the sample, and not a characteristic of the population, you cannot test for it, in the same sense that it makes no sense to test that $\hat \beta = 0$, rather than $\beta = 0$. Of course, there are tests for relationships in the population that are often misinterpreted as test of multicollinearity.